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putting 12 matchsticks end to end, exactly 3 different triangles can be made, what is the formula needed to calculate the number of triangles can...

There are three distinct triangles using 12 matchsticks: 3-4-5,4-4-4, and 5-5-2.

Note that for a given number of matchsticks M, if the sides of the triangle are labeled a,b, and c (without loss of generality let c be the longest side) then a+b+c=M, a+b>c and c<M/2.


We can generate the number of triangles for a given M:


For 24 matchsticks the possible triangles are 11-11-2,11-10-3,11-9-4,11-8-5,11-7-6,10-10-4,10-9-5,10-8-6,10-7-7,9-9-6,9-8-7,8-8-8 or 12 possible triangles.


There are 19 possible triangles for M=30.


For M>=3 then (M,number of triangles) is (3,1),(4,0),(5,1),(6,1),(7,2),(8,1),(9,3),(10,2),(11,4),(12,3),(13,5),(14,4),(15,7),(16,5),(17,8),(18,7)...


There is a formula for this:


If M is even then the number of triangles is the integer closest to M^2/48.


If M is odd then the number of triangles is the integer closest to (M+3)^2/48


For example, if M=12 we have 12^2/48=3.


If M=14 we have 14^2/48=4.083333 so there are 4 triangles.


If M=9 we have (9+3)^2/48=3.


If M=11 we have (11+3)^2/48=4.083333 so there are 4 triangles.


Please see the reference link for a proof.

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